Recent observations of soft gamma-ray repeaters and
anomalous X-ray pulsars are furnishing greater evidence for
the existence of a class of neutron stars with surface
magnetic fields exceeding the critical field of 4.4 x
1013 Gauss. The main effort of this study is to
understand the role of relativistic Compton scattering as it
operates with other QED processes, to develop approximate
expressions of the exact rate and incorporate them in a full
acceleration-cascade model. Previous studies have assumed
that Compton scattering can be adequately described by the
nonrelativistic Compton scattering cross section (Thomson
limit) below resonance and at the resonance or the
Klein-Nishina cross section for nonresonant scattering.
Consequently, these studies have not included the effects of
the strong magnetic fields. The study of the strong field
effects on the inverse Compton scattering process will
provide insight into the particle-photon interactions
associated with a variety of pulsar phenomena, burst spectra
of soft gamma-ray repeaters and the transport of thermal
radiation through neutron star atmospheres. This paper will
summarize the evidence for high-field neutron stars, the
role of inverse Compton scattering and the effects of strong
magnetic fields upon the Compton scattering cross section.